1

Estimation of Intensity Duration Frequency Curves

for Current and Future Climates

By Nicolas Desramaut

Department of Civil Engineering and Applied Mechanics McGill University

Montreal, Quebec, Canada

A thesis submitted to the Graduate and Postdoctoral Studies Office in partial fulfilments of requirements of the degree of Master of Engineering

June 2008

Nicolas Desramaut, 2008

All Rights reserved

2

Acknowledgement

I would like to kindly and very sincerely thank my supervisor, Professor Van-

Thanh-Van Nguyen, for his never-ending help, his indefectible encouragements,

and his professional guidance.

I would also thank Dr Tan-Danh Nguyen and Sbastien Gagnon for their

assistance in my research work, Gilles Rivard and Pierre Dupuis, for their help

using the softwares, the occupants of the room 391 (Reena, Ali, Arden, Damien,

John, Nabil, Reza, Salman) for their friendship and their permanent support, with

a special mention to Arden, who helped me submitting this thesis despite the

ocean.

And last but not least, I would like to express my deeper thanks to my family, who

despite the distance, were always on my side during my stay in Montreal.

i

Abstract

Climate variability and change are expected to have important impacts on the

hydrologic cycle at different temporal and spatial scales In order to build long-

lasting drainage systems, civil engineers and urban planners should take into

account these potential impacts in their hydrological simulations. However, even

if Global Climate Models (GCM) are able to describe the large-scale features of

the climate reasonably well, their coarse spatial and temporal resolutions prevent

their outputs from being used directly in impact assessment models at regional or

local scales.

This study proposes a statistical downscaling approach, based on the scale

invariance concept, to incorporate GCM outputs in the derivation of Intensity-

Duration-Frequency (IDF) curves and the estimation of urban design storms for

current and future climates under different climate change scenarios. The

estimated design storms were then used in the estimations of runoff peaks and

volumes for urban watersheds of different shapes and different levels of surface

imperviousness using the popular Storm Water Management Model (SWMM).

Finally, a regional analysis was performed to estimate the scaling parameters of

extreme rainfall processes for locations with limited or without data. In summary,

results of an illustrative application of the proposed statistical downscaling

approach using rainfall data available in Quebec (Canada) have indicated that it is

feasible to estimate the IDF relations and the resulting design storms and runoff

characteristics for current and future climates in consideration of GCM-based

climate change scenarios. Furthermore, based on the proposed regional analysis of

the scaling properties of extreme rainfalls in Singapore, it has been demonstrated

that it is feasible to estimate the IDF curves for partially-gaged or ungaged sites.

ii

Rsum

La variabilit et les changements climatiques devraient avoir des impacts

considrables sur le cycle hydrologique aux diffrentes chelles spatio-

temporelles. Afin de construire des systmes de drainage durables, les ingnieurs

se doivent de prendre en considration ses modifications probables dans leur

simulation. Toutefois, si les Modles de Circulation Globale (MCG) sont capables

de reproduire raisonnablement bien les caractristiques grande chelle du climat,

leurs rsolutions sont trop grossires pour permettre une utilisation immdiate de

leurs informations dans les modlisations urbaines.

Cette tude propose une approche de mise lchelle statistique, base sur les

proprits dinvariance dchelle, afin dincorporer les rsultats des MCG dans la

conception des courbes Intensit-Dure-Frquence (IDF) futures. Ces courbes

sont ensuite utilises pour lestimation de lvolution des quantits de

ruisslement. Enfin, une analyse rgionale permet une valuation des paramtres

de rduction dchelle pour des stations partiellement jauges, voir non jauges.

Cest ainsi que des courbes IDF peuvent tre construites avec un nombre limit de

donnes.

iii

Table of Content

Table of Content .................................................................................................................. iii List of Figures ..................................................................................................................... iv 1. Introduction ................................................................................................................. 1

1.1. Context............................................................................................................... 1 1.2. Research Objectives .......................................................................................... 3 1.3. Literature Review ............................................................................................... 4

2. Evaluation of future IDF curves and runoff values in Quebec assuming stationarity in the scaling properties .......................................................................................................... 7

2.1. Data ................................................................................................................... 7 2.2. Methodology .................................................................................................... 13 2.3. Results and Discussion ................................................................................... 20 2.4. Validation ......................................................................................................... 26 2.5. Future Runoff estimations................................................................................ 33 2.6. Discussion ....................................................................................................... 35 2.7. Future researchs .............................................................................................. 35

3. Regional analysis of rainfall scaling properties ........................................................ 36 3.1. Introduction ...................................................................................................... 36 3.2. Data ................................................................................................................. 36 3.3. Methodology .................................................................................................... 43 3.4. Results ............................................................................................................. 44 3.5. Discussion ....................................................................................................... 50

4. Conclusions ............................................................................................................. 51 Bibliography ....................................................................................................................... 52 APPENDICES ................................................................................................................... 56 A - Analysis of the consistency between the different sets of data ............................... 56 B - Computation of GEV Quantile ................................................................................. 57 C - Determination of the three GEV parameters from the 3 first NCM ......................... 58 D - Quantile Plots of the AMP for the baseline period .................................................. 61 E - Evolution of the design storms ................................................................................ 66 F - Evolution of the runoff values in the different watershed configurations ................. 70

iv

List of Figures

Figure 1: Localization of the Pierre-Elliot-Trudeau International Airport in Quebec ........ 7 Figure 2: Example of a recorded historical storm ............................................................... 8 Figure 3: Difference between raw SDSM estimates and observed values for the 1961-1994 baseline period ......................................................................................................... 10 Figure 4: Adjustment functions for daily GCM downscaled AMP for the 1961-1994 baseline period .................................................................................................................. 10 Figure 5: Box plots of the 100 adjusted AM Daily Rainfall estimations from CGCM2A2 compared with the observed values (dots), for the 1961-1994 period. ............................. 11 Figure 6: Box plots of the 100 adjusted AM Daily Rainfall estimations from HadCM3A2 compared with the observed values (dots), for the 1961-1994 period. ............................. 11 Figure 7: Difference between observed AMP and recorded historical storms.................. 12 Figure 8: Empirical IDF Curve, baseline period (1961-1994) .......................................... 14 Figure 9: Example of a Desbordes Design Storm ............................................................. 18 Figure 10: Example of a Peyron Design Storm ................................................................ 20 Figure 11: Example of a Watt et al. Design Storm ........................................................... 20 Figure 12: Quantile plots of the 1hr AMP for the baseline period .................................... 21 Figure 13: linearity of the NCM (AMPs, baseline period 1961-1994) ............................. 21 Figure 14: Linearity of the scaling slopes (*k) ............................................................... 22 Figure 15: IDF Curves built with scaling concept for the baseline period symbols= observed values and line = Scaling GEV model ............................................................... 23 Figure 16: three design storms from the observed data, baseline period 1961-1994 ........ 24 Figure 17: Volume estimations from the three design storms based on observed data, baseline period 1961-1994 ................................................................................................ 25 Figure 18: Peak flow estimation, from the three design storms based on observed data, baseline period 1961-1994 ................................................................................................ 25 Figure 19: IDF and design storms conceived from SDSM driven by CGCM2 predictors for 1980-1994, with parameters calibrated over the 1961-1979 period ............................ 26 Figure 20: IDF and design storms conceived from SDSM driven by HadCM3 for 1980-1994, with parameters calibrated over the 1961-1979 period ........................................... 27 Figure 21: Simulated volumes from the Observed GCMs using Watt design storm model vs volume from the historical storms ................................................................................ 27 Figure 22: simulated peak flows from the GCMs using Desbordes design storm models vs volume from the observed storms ..................................................................................... 28 Figure 23: IDF Curves generated from SDSM-CGCM2unbiased daily precipitation, for current and future periods ................................................................................................. 30 Figure 24: Evolution of the daily intensity according to SDSM-CGCM2 with respect to the the baseline period....................................................................................................... 30 Figure 25: IDF Curves generated from HadCM3, for current and future periods ............ 31 Figure 26: Evolution of the daily intensity according to HadCM3 in comparison with the baseline period .................................................................................................................. 31 Figure 27: Design storms for the current and future periods, with a 2-year return period, from both SDSM-GCM. ................................................................................................... 32 Figure 28: Evolution of the runoff peak flows (Watershed, square, 65%, 1ha) ................ 33 Figure 29: Evolution of the runoff volume, (Watershed, square, 65%, 1ha) .................... 34 Figure 30: Spatial repartition of the 9 stations in Singapour ............................................ 38 Figure 31: NCM vs duration to test the scaling behaviour of the AMP in the 9 Singapore stations .............................................................................................................................. 40 Figure 32: Proportionality of the scaling slopes in the 9 Singapore stations .................... 41

v

Figure 33: IDF Curves for Ama Keng station derived using Scaling GEV invariance .... 42 Figure 34: daily AM vs. scaling slopes ............................................................................. 44 Figure 35: mean number of rainy days (RD) in January vs. scaling slopes ...................... 45 Figure 36: mean number of rainy days (RD) in February vs. scaling slopes .................... 46 Figure 37: Results of the jackknife method with daily AMP as rainfall parameters for partially gauged site. ......................................................................................................... 47 Figure 38: Results of the jackknife method with number of rainy days in January as rainfall parameters for partially gauged site. ..................................................................... 47 Figure 39: Best result with rainfall parameter =January rainy Day for the Jackknife method for the 9 stations ................................................................................................... 50 Figure 40: Design Storms for the 4 periods for the 2-year return period .......................... 66 Figure 41: Design storms for the 4 periods for the 5-year return period .......................... 67 Figure 42: Design storms for the 4 periods for the 10-year return period ........................ 68 Figure 43: Design storms for the 4 periods for the 50-year return period ........................ 69 Figure 44: Evolution of the runoff peak flows (Rect, 100%, 0.4ha) ................................. 70 Figure 45: Evolution of the runoff peak flows (Rect, 100%, 2 ha) ................................... 71 Figure 46: Evolution of the runoff peak flows (Square, 65%, 10 ha) ............................... 72 Figure 47: Evolution of the runoff volume (Rect, 100%, 0.4 ha) ..................................... 73 Figure 48: Evolution of the runoff volumes (Rect, 100%, 2ha) ........................................ 74 Figure 49: Evolution of the runoff volumes (Square, 65%, 10 ha) ................................... 75

vi

List of Tables

Table 1: Repartition of the historical storms during the 1961-1994 baseline period .......... 9 Table 2: Watershed configurations ................................................................................... 16 Table 3: Difference between volumes generated from the observed storms and from the simulated values of design storms .................................................................................... 25 Table 4: Difference between peak flows generated from the observed storms and from the simulated values of design storms .................................................................................... 25 Table 5: Difference (in %) between volumes generated from the observed storms and from the different models of design storms according to respective return periods ......... 28 Table 6: Difference (in %) between peak flows generated from the observed storms and from the different models of design storms according to the respective return periods. .. 28 Table 7: Periods of data availability for daily rainfall and 8 durations for AMP.............. 37 Table 8 : Months of occurrence of the annual maximum daily rainfall ............................ 39 Table 9: Scaling GEV parameters for the 9 Singaporean stations .................................... 42 Table 10: Correlation coefficient between scaling slopes and hydrologic values. ........... 45 Table 11: performance criteria (R) of the jackknife method for partially gauged stations, for the 14 climatic parameters ........................................................................................... 46 Table 12: Performance of the parameter estimation method (com=computed values and obs=observed values, the percentage being the difference between them) ....................... 48 Table 13: performance criteria (R) of the jackknife method for un-gauged stations. ...... 49 Table 14: Relative difference between observed AMP and annual maximum precipitation depths computed from the available historical storms ...................................................... 56

1

1. Introduction

1.1. Context

Historically, the first urban drainage systems were built to protect the urban

infrastructures from water damage. Their main task was then to convey

stormwater runoff outside the city as quickly as possible. The opportunity to use

these systems to dispose of wastewater appeared later, in the XIXth century, with

the raising awareness of the impact of polluted water on health. Hence, the

current engineering practice for the design of urban drainage systems requires

taking into account these two main objectives: the removal of wastewater and the

drainage of storm runoff. The first objective could be achieved by considering

different wastewater treatment measures to prevent the discharge of polluted

water into the receiving environment. The second objective is more difficult to

address because of the random variability of runoff in time and in space due to the

randomness of precipitation for current and future climates. Furthermore, the

runoff process could be non-homogeneous due to expanding urbanization of the

watershed causing changes in land use pattern over time (e.g., natural drainage

systems such as streams have been replaced by impervious areas such as streets

and residential buildings). Therefore, in addition to the effects of land use changes

on the runoff process, improved estimations of precipitation patterns for current

and future climates are urgently needed to provide more accurate estimation of

runoff characteristics in the context of a changing climate.

Most urban drainage systems were built to last for a long time, usually more than

50 years, because their repair or retrofitting could be expensive and could cause

massive disruptions of normal city activities since these systems are mainly

located in underground areas. These infrastructures are designed to cope with the

majority (but not all) of the extreme rainfall events that could occur during their

service life (usually ranging from 50 to 100 years). In fact, in view of economic

but also technical constraints, there are some limits to the capacities of drainage

2

systems. The current practice has set some criteria for the construction of drainage

systems to balance construction constraints with levels of risk. The minor

drainage system (piped system) should be built to be overwhelmed only every 5 to

10 years, whereas the major drainage system (overland system, including the

roads and parking lots) should support nearly every event (Arisz and Burrell,

2006).

To meet the above-mentioned criteria, engineers need accurate estimations of

potential runoffs from an urban watershed. However, historical runoff data are

generally lacking or have become irrelevant due to changes in land use. In order

to cope with this deficiency, engineers rely on conceptual urban runoff models,

such as the SWMM model, to simulate the rainfall-runoff relations. Indeed, these

models could describe the complex interactions between different hydrologic and

hydraulic processes involved in an urban watershed to transform the input

precipitation pattern into the complete output runoff hydrograph. Hence, to use

these models, knowledge of current and future rainfall patterns, generally

presented as standardized hyetographs, is required. This synthetic pattern of

precipitation provides illustrations of probable distributions of rainfall intensity

over time for a given return period at a local site. In general, these synthetic

hyetographs are computed based on historic rainfall records, with the assumption

that the distributions parameters are stationary or constant over time.

However, recent studies compiled in the Intergovernmental Panel on Climate

Change assessment report (IPCC, 2007) have shown that extreme rainfall events

are very likely to be globally more frequent and more intense in future decades.

Thus, the hypothesis of stationary precipitation occurrences appears highly

questionable in prospect of climate change. Hence, the design storms computed

based on historic data could only be accurate for the current period but not for the

whole service life of drainage systems. Consequently, to build sustainable

drainage systems, engineers need to incorporate future climatic conditions into

3

their computation of design storms. The Global Climate Models (GCMs), also

called General Circulation Models, are considered to be able to provide reliable

scenarios for future climatic parameters at the global scale. However, their

resolutions are too coarse (~300 km, 1 day) for being able to be directly used in

the simulation of runoff from urban watersheds since these watersheds are

normally characterized by a short response time (usually, less than 1 day) due to

small surface area size and high imperviousness level. Therefore, outputs from

GCMs should be downscaled to provide climate simulations at appropriate

temporal and spatial scales that are required for accurate estimation of rainfall and

runoff characteristics for urban watersheds.

1.2. Research Objectives

In view of the above-mentioned issues, the present study proposes a new

statistical downscaling method for estimating the Intensity-Duration-Frequency

(IDF) relations that could take into account the climate information given by

GCMs for current as well as for future climatic conditions. More specifically, this

study is aiming at the following particular objectives:

To propose a spatial-temporal statistical downscaling method to describe

the linkage between daily global climate variables and local daily and sub-daily

extreme rainfalls for the derivation of IDF relations for current and future

climates;

To propose a procedure for constructing IDF relations for a local site that

could take into account GCM-based climate change projections;

To propose a procedure for estimating the urban design storms in consideration of

GCM-based climate change scenarios;

To propose a procedure for evaluating the impacts of climate change on

the runoff process for urban watersheds; and to investigate the spatial variability

of extreme rainfall scaling properties over a given region.

4

The first part of the study was performed assuming stationary scaling parameters

for extreme rainfalls (i.e., the spatial and temporal scaling properties for future

periods will be the same as the ones found for the current period). The main aim

is to assess the future trends in extreme rainfall events and their consequences on

design rainfalls and urban runoff properties for urban drainage system design.

Notice that the hypothesis of stationary rainfall scaling parameters could only

represent an approximation since these values could change due to the change in

climatic conditions for future periods.

In order to deal with this stationarity issue as well as with the situations where

rainfall records at the site of interest are limited or unavailable, a method is

proposed in the second part to examine the spatial variability of the rainfall

scaling parameters in terms of the regional variation of some climatic variables.

This regional analysis could permit the estimation of IDF curves for current

periods for cases with limited or without data using records from surrounding

stations as well as for the estimation of IDF relations for future periods using

GCM-based projections to avoid the stationarity hypothesis.

1.3. Literature Review

1.3.1. Consequences of Climate Change on Urban Drainage

During the last decades, scientists have become more and more conscious of the

potential impacts of climate change on precipitation patterns for the design of

urban drainage systems. Most previous studies have agreed upon the fact that, as

rainfall events are likely to become more frequent and more intense, the level of

services provided by the drainage systems will be reduced and these urban

structures will suffer more frequent damage, reducing their service life. However,

as Semadeni-Davies (2004) points out, there are neither tools nor guidelines in the

5

literature to assess these impacts of climate change in urban catchment areas. In

view of this deficiency, this study proposes the uses of incremental scenarios,

with increases of precipitation arbitrarily taken in a range of the values generated

by the GCM. This method is simple and provides a range of possible

consequences. However, as recognized in the study , these scenarios should not be

considered as future climatic conditions, but only as a spectrum of potential

impacts, as a result of the arbitrary forcing. . Watt et al. (2003) used a similar

approach for estimating the evolution of the runoff values. They assumed that the

rainfall intensity was increased by 15 percent. From the resulting runoff values,

they estimated the parts of the drainage systems that will face overflow capacity.

However, this method is based on crude estimates. They assumed that intensities

of all rainfall events evolved in a similar trend, independently of their duration,

with an increase directly estimated from the GCM large-scale trends. Hence, this

method may not be suitable for urban catchment, where the change in

precipitation patterns might be local but not uniformly proportional.

Denault et al. (2002) proposed an alternative to these methods. They

determined the past linear trends of local rainfall depth for different durations

ranging from 5 min to 1 day. Based on this information, and on the assumption

that the linear increases remain constant, they extrapolated the future intensity-

duration relationships, and, from these data, construct the corresponding design

storms for future periods. They use the software SWMM (Huber and Dickinson,

2002) to perform estimations of future runoff values. This method is an alternative

for assessing potential variations of design storms resulting from climate change,

and therefore for evaluating hydraulic responses of catchment areas, taking into

account that the shape of the design storms can be altered. However, the authors

have to make the hypothesis that the increase rate of the local rainfall depth will

remain the same (i.e. the increase will continue to be linear). The outputs of the

GCM, usually without any persistent linear tendency, make this hypothesis highly

questionable. Moreover, the recent IPCC report (IPCC, 2007) suggests that the

6

evolution of anthropogenic activities is very likely to have significant impacts on

future climate conditions. And this evolution would certainly not be linear. So we

need to take into account uncertain nonlinear future causes to assess future effects.

Thus, even if scientists and decision-makers point out the need for a clear

understanding of the effects of climate change on human and social activities,

there is no agreement regarding the suitable approach for evaluating its impacts

on urban hydrologic processes.

1.3.2. Design Storms

Design storms have been used extensively by civil engineers to size sewers for

more than 60 years. With the nonlinear interactions between rainfall and runoff

processes as described by various urban runoff models, these synthetic design

storms are required for the estimation of the complete runoff hydrographs for

urban drainage design purposes. Many frameworks have been conceived in

different countries for the computation of design storms (Marsalek and Watt,

1984), with varying shapes, storm durations, time to peak, maximum intensity and

total volume of rainfall; however, none matches every situation, forcing

hydrologists to perform assessment processes before using a design-storm model

at a new site (Peyron, 2001). Peyron et al. (2005) proposed a procedure to

systematically evaluate design storms models for specific locations. First, the

targeted design storms for different return periods, based on rainfall data from

local Intensity-Duration-Frequency (IDF) curves, were constructed. Then, these

synthetic hyetographs were used as input in the SWMM model to obtain the

respective runoff values (peak flows and volumes). Thus, the runoff properties

estimated from the design storms were compared to those values obtained from

observed historical storms to assess the accuracy of different design storm models.

Based on this approach, the best design storm can be selected for the design of

urban drainage systems.

2. Eva

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8

3.1.2. Observed data

The observed Annual Maximum Precipitation (AMP) data available at the Pierre

Elliot Trudeau airport for the 1961-1990 period were used in the present study.

The AMP data (representing the maximum rainfall depth fallen each year on a

daily or sub-daily basis) measured in millimetres (within a precision of one-tenth

of millimetres) and available for 9 durations (5, 10, 15, 30 minutes; 1, 2, 6, 12 and

24 hours) were provided by Environment Canada.

Figure 2: Example of a recorded historical storm

To investigate the accuracy of the design storm models, 140 historical rainfall

events (called storms in the remaining of the report ) were recorded during this

1961-1994 baseline period, in millimetres, with a 5-minute time step and with a

one-tenth-millimetre precision. An example of a recorded storm is given in Figure

2. The selection was performed according to their ability to generate the biggest

runoff events and the presence of at least one recorded storm event per year.

0 0 03,1

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9

Year 1961 1962 1963 1964 1965 1966 1967 1968 1969 # of storms 5 2 5 2 2 4 2 1 1

Year 1970 1971 1972 1973 1974 1975 1976 1977 1978 # of storms 1 2 3 5 5 7 5 3 3

Year 1979 1980 1981 1982 1983 1984 1985 1986 1987 # of storms 1 6 3 1 2 4 1 8 8

Year 1988 1989 1990 1991 1992 1993 1994 # of storms 4 5 11 2 3 7 16

Table 1: Repartition of the historical storms during the 1961-1994 baseline period

3.1.3. Simulated data

Daily rainfalls for the baseline 1961-1994 period and for three 30-year future

periods (2010-2039; 2040-2069; and 2070-2099) were provided by Dr. Tan-Danh

Nguyen (Department of Civil Engineering and Applied Mechanics, McGill

University) using the Statistical Downscaling Model (SDSM) (Wilby et al. 2002),

from driven conditions from Global Climate Models (GCM) inputs (Nguyen et

al., 2006). This linear regression based downscaling method allows the

development of the linear statistical linkages between large-scale predictors from

GCMs and local observed predictands (e.g., the daily rainfall in this study). The

GCM climatic variable data are from two different GCMs:

Hadley Centre Coupled Model, version 3 (HadCM3) (Johns et al., 2003), Hadley Centre, United Kingdom

Coupled Global Climate Model, second generation (CGCM2), (Flato and Boer, 2001), Canadian Center for Climate modelling and analysis, Canada

These series of simulations under climate change conditions use the SRES A2

emission scenario with the hypothesis of greenhouse gases (GHG) emissions

following the SRES A2 scenario (Nakicenovic et al., 2000). This scenario is based

on the assumption that the world will become more and more heterogeneous, with

a continuously increasing population (15 billion by 2100), slow and regional

economic growths, and few inter-country technological transfers. This scenario

forecasts that the level of CO2 in the atmosphere will be double in 2100 compared

with the pre-industrial concentration. In the present study, a set of 100 simulations

were performed for each model and for each future period considered.

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11

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12

3.1.5. Selection of years with data inconsistency

In order to compare in a suitable manner the outputs of the computations with the

observed data , the different sets of data have to be consistent. The maximum

depths of precipitation resulting from the available recorded historical storms for

8 durations (5, 10, 15 and 30 minutes; 1, 2, 6 and 12 hours) are computed for each

year, and compared with the corresponding recorded AMP (APPENDICE A).

Three years (1970, 1985 and 1991) have inconsistent sets of data, as illustrated in

Figure 7 with the average of the relative difference over the 8 durations. This

inconsistency is due to the absence of the record of the biggest historical storm in

1970, 1985, and 1991 in our data sets. However, for these 3 years, the different

values were near the median value for each duration. Hence, the removal of these

data should not trigger significant bias in the simulations.

Figure 7: Difference between observed AMP and recorded historical storms

0%

100%

200%

300%

400%

500%

600%

1961

1964

1967

1970

1973

1976

1979

1982

1985

1988

1991

1994

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ce(%

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Year

13

3.2. Methodology

3.2.1. Construction of IDF Curves

3.2.1.1. Generalized extreme values distribution

The General Extreme Values (GEV) distribution is a distribution commonly used

to model extreme events, in particular extreme precipitations. Indeed, it has been

proven to match well the observed extreme values (Nguyen et al., 1998). The

GEV cumulative distribution function can be written as

; , , 1

(1)

where X is the quantile associated with a return period ; , and are

parameters for respectively the location, the scale and the shape.

As a result, the quantile X for the return period can be calculated as follow:

/ ^ (2)

in which p is the probability of exceedance, related to the return period:

/ (3)

3.2.1.2. GEV parameters estimations

The three GEV parameters are estimated by the method of the Non Central

Moments (NCM), where the NCM are defined as:

(4)

It can be demonstrated (Nguyen et al. (2002)) that they are related to the NCM

according to these equations.

(5)

in which (.) is the gamma function

14

Thus, using the first three NCMs (k = 1, 2, and 3) given by Equation (4) the three

parameters of the GEV distribution , and can be determined by the method of

moments (see APPENDICE B)

3.2.1.3. Empirical IDF curves

The AMPs were ranked in descending order to compute the exceedance empirical

probability pi for each rank i, using the Cunnanes plotting position formula

(Cunnane, 1978):

.

. (6)

in which n is the number of years of the record.

Then the return periods, assuming that there are no trends in the quantiles values

(i.e., p is invariant over time), are computed as:

(7)

Thus, the empirical frequencies (or return periods) of AMPs can be estimated for

different rainfall durations, and inversely, the AMP quantiles for a given duration

can be computed for specific return periods. The empirical IDF curves can hence

be plotted as shown in Figure 8.

Figure 8: Empirical IDF curves for the 1961-1994 baseline period

15

3.2.1.4. Temporal downscaling

There are empirical evidences that some characteristics of short-duration rainfalls

can be deduced from longer duration extreme events (Over and Gupta, 1994,

Aronica and Freni, 2005; and Nhat et al., 2006). Indeed, extreme precipitation

events often present scale invariance properties (Burlando and Rosso, 1996). This

scaling concept can be expressed as:

If a function f is scaling invariant, there is a constant such that: ,

(8)

Nguyen et al. (2002) proved that such functions can be formulated as:

, (9)

Hence, the NCM, for a duration t, can be presented in a similar formulation:

, (10)

(11)

(12)

As a consequence, the three GEV parameters and the quantile X for sub-daily

durations can be computed from daily GEV parameters using these properties

(Nguyen et al., 2006):

(13)

(14)

(15)

(16)

It should be noticed that precipitations can present different scaling regimes. In

other words, they can exhibit scale invariance properties, with different scaling

parameters (i.e. different ) during distinct duration intervals (called scale

16

interval). For example, if a function has two scaling regimes, one between t1 and

t2, and the other one between t2 and t3, the function can be described as:

;

;

(17)

Thus, the IDF curves can be fully defined with only the three estimated

parameters for the GEV distribution for the daily duration and the scaling

parameters corresponding to the different extreme rainfall scaling regimes.

3.2.2. Estimation of Runoff Volumes and Peak Flows

3.2.2.1. Watershed Configurations

Different typical urban watersheds have been considered in this part. Their

characteristics are listed in Table 2. They represent some of the configurations

(imperviousness, area, shape) representative of cities in Quebec.

TYPE PERCENT OF IMPERVIOUSNESS(%) AREA (ha) Shape

Parking lots 100 % 0.4 and 2 rectangular

High density residential area 65 % 1 and 10 square

Table 2: Watershed configurations

For illustration purposes, only the results of the small residential area

configuration (square, 65%, 1 ha) are presented in the text and the results for the

other watershed configurations are described in Appendice E

17

3.2.2.2. Storm Water Management Model (SWMM)

The Storm Water Management Model (SWMM) developed by the US

Environmental Protection Agency is widely used for the estimation of urban

runoff from rainfall (Huber and Dickinson, 1988). Discrete rainfall hyetographs

are used as input for this model. As the purpose of this study is not to investigate

the performance of the SWMM but to assess the ability of the proposed

downscaling method to mimic actual precipitation pattern and to estimate the

evolution of runoff characteristics due to change in precipitation patterns; hence,

no specific calibration of the SWMM is necessary. Indeed, comparisons between

runoff properties computed from synthetic design storms were compared with

those given by the observed real storms to assess the accuracy of the suggested

estimation method.

In the first step, the runoff values generated by the 136 historical storms were

first computed. Annual maximum volumes and peak flows were extracted from

the results and a frequency analysis of these two extreme series was carried out.

The Gumbel distribution (Extreme Value Type I) provides the best fit for these

runoff value (Mailhot et al., 2007b). Thus the empirical quantiles for the runoff

values can be expressed as:

(18)

. (19)

where is the mean of the runoff volume (or peak flow) and the standard

deviation.

In the second step, synthetic hyetographs for current and future periods were used

to validate the computed runoff for the current period and to make the projection

of runoff for future periods. These synthetic hyetographs are called (synthetic)

design storms.

18

3.2.2.3. Design Storms

3.2.2.3.1. General consideration

Peyron (2001) has demonstrated that three design storm models proposed by

Desbordes (1978), Watt et al. (1986), and Peyron et al. (2001) could provide

runoff properties that are similar to those given by the observed historical storms.

These three design storm models were thus chosen for this study. More

specifically, the selected design storms were computed for different return periods

based on the computed IDF curves for Dorval Airport as described previously. In

addition, all these synthetic storms have 1-hour duration with a discrete time step

of 5 minutes.

3.2.2.3.2. Desbordes Design Storm

Figure 9 shows the shape of the Desbordes design storm. The maximum intensity

is located at the 25th minute of the 1-hour storm duration.

Notation:

Equation

;

;

;

;

Fig

Fi

Figu

gure 9: Exam

igure 10: Exa

ure 11: Exam

19

mple of a Desb

ample of a Pe

mple of a Wat

bordes Design

eyron Design

tt et al. Desig

n Storm

Storm

gn Storm

20

3.2.2.3.3. Peyron Design Storm

The period of maximal intensity lasts 15 minutes between the 20th and 35th

minutes and the intensity peak occurs at the 25th minute (Figure 10). The intensity

is constant outside of the interval. This design storm generates a total depth 1.3

times higher than the actual 1-hour maximal volume for the same return period.

Equation:

. . .

, ; ;

. , ; ; . , ;

3.2.2.3.4. Watt Design Storm

The maximum of intensity takes place in the 5th time interval (Figure 11).

Equation:

, ;

, ;

3.3. Results and Discussion

3.3.1. Verification and Validation Process

In order to assess the feasibility of the proposed downscaling procedure to

generate accurate runoff properties for the current and future periods. The

accuracy of the scaling GEV distribution in the estimation of AMPs for different

durations is evaluated.

3.3.1.1. Accuracy of the Scaling GEV

In order to assert the ability of the GEV distribution to fit observed data, quantile

plots were drawn for each of the 9 durations (Figure 12 and APPENDICES D).

The goodness-of-fit found for all the durations corroborates the hypothesis of

AMPs following GEV distributions.

Figu

The scaling

plotting the

the duration

scaling func

So

Thus, if the

proportiona

Figu

ure 12: Quan

g behaviou

e logarithm

n. Indeed,

ction could

ktE

lnkt

e function i

al with a fac

ure 13: Linear

ntile plots of t

ur of the A

of the NCM

as previous

be written

Efk1tk

lnEfk1

s scaling, th

ctor .

rity of the N

21

the 1hr AMP

AMP record

M from the

sly demonst

as

1klnthe plot shou

CM of AMPs

for the basel

ded at the

AMP serie

trated, the n

t

uld be linea

s for the base

line 1961-199

station is i

es, against th

non central

ar and the s

eline 1961-199

94 period

investigated

he logarithm

l moments

(20)

(21)

lopes would

94 period

d by

m of

of a

d be

Hence, the

indicates th

are two di

(1=0.50) a

Based on th

correspond

derived fro

the daily A

were comp

Figure 15,

ones (symb

provide acc

e linearity o

he scaling b

ifferent sca

and the seco

Fi

he two estim

ing GEV d

om the thre

AMP. On t

puted, and

the estimat

bols) (R~0

curate descr

of the grap

behaviour of

aling regim

ond between

igure 14: Lin

mated scalin

distribution

ee NCMs an

the basis o

the resultin

ed IDF cur

.99). Hence

ription of th

22

phs exhibit

f the AMPs

mes: the fir

n 30 minute

earity of the

ng paramet

parameters

nd the assoc

of these GE

ng IDF cur

rves (lines)

e, the propo

he AMP dist

ed on Figu

s at Dorval

rst one bet

s and 1 day

scaling slope

ers 1 and

s for the 8

ciated comp

EV distribu

rves were

agreed ve

osed NCM

tributions.

ure 13 and

airport. In p

tween 5 an

y (2=0.21).

s (*k)

2, the three

sub-daily

puted GEV

utions, the

derived. A

ery well wit

M/scaling GE

d on Figure

particular, t

nd 30 min

e NCMs and

durations w

parameters

AMP quan

As indicated

th the empir

EV method

e 14

here

nutes

d the

were

s for

ntiles

d by

rical

can

23

Figure 15: IDF Curves built with scaling concept for the baseline period symbols= observed values and line = Scaling GEV model

3.3.1.2. Estimation of runoff properties

The three different design storms (Figure 16) were derived using the computed

IDF curves as described above. These design storms were then used as input to

the SWMM model to estimate the corresponding runoff peaks and volumes for the

selected urban watersheds. Similarly, these two runoff properties were computed

for the historical storms using the SWMM simulation with the same conditions as

for the synthetic design storms.

As expected, the Watt design storm model gave the best performance in the

estimation of the runoff volumes, while the (Figure 17 and Tables 3 and 4)

Desbordes model provided an accurate estimation of the peak flows (Figure 18

and Table 4), and the Peyron model gave accurate estimations for both of these

runoff properties.

Figure 16: TThree designn storms from

24 m the observeed data for thhe 1961-1994

4 baseline perriod

Tab

Tab

Figure 17: Eand from

Volume Desbordes

Peyron Watt et al. ble 3: Differe

Figure 18: E and from

peak flow Desbordes

Peyron Watt et al. le 4: Differen

Estimation ofm observed st

2 years -17,35% -9,57% 4,21%

ence betweenand f

Estimation ofm observed st

2 years 4,51% 2,46%

35,25% nce between p

and f

25

f runoff volumtorms for the

5 years-16,21%-8,19%4,93%

volumes genfrom the desi

f peak flows torms for the

5 years-3,56%-8,14%23,40%

peak flows gefrom the desi

mes from thr1961-1994 b

10 yea% -15,29%

-7,14%5,59%

nerated from ign storms

from the thr1961-1994 b

10 yea-6,64%-10,01%19,61%

enerated fromign storms

ree design storaseline period

rs 50 ye% -13,4

% -5,23% 6,95

the observed

ee design storaseline period

rs 50 ye% -8,90% -12,3% 14,75m the observe

rms d

ears 40% 3% 5% d storms

rms d

ears 0% 39% 5%

ed storms

26

3.4. Validation

The downscaled daily rainfall series were generated using the calibrated SDSM

procedure for two GCMs (the UK HadCM3 and the Canadian CGCM2) for the

1961-1979 period. The design storms and the resulting runoff peak flows and

volumes were then estimated based on the 100 generated downscaled daily

rainfall time series and the bias-correction procedure as suggested in section 2.1.4.

for the 1980-1994 period (Figure 19 and 20). The computed runoff values from

design storms were compared with those from historical storms for the 1980-1994

period.

Figure 19: IDF and design storms estimated based on SDSM downscaling results using CGCM2 predictors for 1980-1994, with parameters calibrated over the 1961-1979 period

Comparable results were found for runoff values computed for these two cases for

both HadCM3 and CGCM2 models (Tables 5 and 6, Figures 21 and 22). The

differences range from 0.5 to 10% for the runoff volumes using Watt model and

between 0.6 and 11.5 % for the peak flows using Desbordes model. Hence, the

27

proposed procedure is able to produce consistent IDF curves and to provide good

estimations of runoff values when an appropriate design storm model was

selected.

Figure 20: IDF and design storms conceived from SDSM driven by HadCM3 for 1980-1994, with parameters calibrated over the 1961-1979 period

Figure 21: Simulated runoff volumes from the GCMs using Watt design storm model as compared with the volume from the historical storms (diamond= CGCM2 and square

HadCM3)

0

200

400

600

0 200 400 600

simulated

volum

e(m

3)

observedvolume(m3)

28

VOLUMES Return period (years) Desbordes 1,5 2 5 10 20 50 CGCM2 -28,1% -28,1% -26,1% -24,8% -23,6% -22,3% HadCM3 -27,8% -27,2% -24,1% -22,3% -20,8% -19,2%

Peyron 1,5 2 5 10 20 50

CGCM2 -21,9% -21,7% -19,4% -17,9% -16,7% -15,4% HadCM3 -21,5% -20,7% -17,2% -15,2% -13,6% -12,0%

Watt et al. 1,5 2 5 10 20 50 CGCM2 -9,1% -8,5% -6,5% -5,3% -4,3% -3,2% HadCM3 -8,6% -7,3% -4,0% -2,3% -0,9% 0,5%

Table 5: Difference (in %) between volumes generated from the observed storms and from different design storm models for different return periods

Figure 22: Simulated peak flows from the GCMs using Desbordes design storm models as

compared with peakflows from the observed storms(diamond= CGCM2 and square HadCM3)

FLOWS return period (years) Desbordes 1,5 2 5 10 20 50 CGCM2 5,7% -0,8% -7,8% -9,9% -11,1% -11,5% HadCM3 5,7% 0,6% -5,5% -7,3% -7,9% -8,0%

Peyron 1,5 2 5 10 20 50

CGCM2 4,0% -2,8% -11,0% -12,9% -14,0% -14,4% HadCM3 4,0% -1,4% -9,2% -10,3% -10,8% -10,9%

Watt et al. 1,5 2 5 10 20 50 CGCM2 39,2% 30,7% 20,5% 17,4% 15,0% 13,7% HadCM3 39,2% 32,1% 23,7% 20,7% 19,1% 17,9%

Table 6: Difference (in %) between peak flows generated from the observed storms and from different design storm models for different return periods.

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0 0,1 0,2 0,3 0,4

simulated

flow

s(m

3/s)

observedflows(m3/s)

29

3.4.1. Simulations for future periods

3.4.1.1. IDF Curves for future periods

As part of the reliability of the results depends on the length of the calibration

period, the calibration steps take into account all the observed data (i.e. the

calibration period will be the baseline periods, 1961-1994, without the 3 years

with inconsistent data). IDF curves, design storms and runoff estimations are

produced for the baseline period using SDSM-GCM unbiased daily simulations

and the three future periods (2020s, 2050s and 2080s), for both GCM (CGCM2

and HadCM3).

The GCM provides two different trends for future extreme rainfalls (Figures 24

and 26). Even if both predict a small decrease of intensities, for all return periods,

during the first period (2020s), the downscaled runs using the CGCM2

predictors show a significant increase (~ +10%) for the last 2 periods (2050s,

2080s), whereas the HadCM3 indicates a continuously slight decrease (reaching ~

-4% in 2080s). However, these changes are not identical in proportions for the

different return periods. Indeed, according to the results obtained from CGCM2

outputs (Figures 23 and 24), the decrease during the first 30-year period is more

pronounced for the rarest events (i.e. with the biggest return periods), but the

relative increase in one century would be approximately the same for all the return

period. The data from HadCM3 model provides different results (Figures 25 and

26). The small decrease in intensity will be fairly similar between the different

frequencies of events.

Figure 23

Figure 24: E

1intensity(m

m/hr)

: IDF Curves

Evolution of

1

10

00

1

s generated frcurre

the daily intet

10

196119942020s2050s2080s

30

rom SDSM-Cent and futur

ensity accordthe baseline p

100

duratio

CGCM2unbiare periods

ding to SDSMperiod

0

n(min)

CGC

ased daily pre

M-CGCM2 wit

1000

CMA2

ecipitation, fo

th respect to

10000

or

the

Figure

Figure 26:

1

10intensity(m

m/hr)

e 25: IDF Cur

Evolution of

1

10

00

1

rves generate

f the daily int

10

19611994

2020s

2050s

2080s

31

ed from HadC

tensity accordbaseline per

100duratio

4

CM3, for cur

ding to HadCriod

0 1on(min)

H

rrent and futu

CM3 in comp

1000

HadCM3A

ure periods

arison with t

10000

the

Figure 27:

3.4.1.2.

CGCM

Design storm

Design Sto

M2

ms for the curfrom

32

orms

rrent and futm both SDSM

ture periods, M-GCM.

HadCM3

with a 2-year

3

r return perio

od,

33

The design storms are built based on these new curves (e.g. for the 2-year return

period, Figure 27 or Appendices E, Figures 40-43). As expected, the change of

the rainfall intensities are similar to the ones noticed in the IDF curves, with a

slight reduction when the hyetograph are computed based on HadCM3 inputs and

an increase when CGCM2 predictors are used in the downscaling process.

3.5. Future Runoff estimations

SWMM simulations were performed for the different watershed configurations.

Again, the results exhibit different trends according to the GCM selected (Figures

28 and 29). Again, the same conclusions can be drawn: Two different changes are

probable, increase according to CGCM2 (despite the initial decrease) and slight

decrease with HadCM3.

CGCM2 HadCM3

Figure 28: Evolution of the runoff peak flows (Watershed: square, 65%, 1ha)

34

CGCM2 HadCM3

Figure 29: Evolution of the runoff volume, (Watershed: square, 65%, 1ha)

Model : Peyron

0

200

400

600

800

2 5 10 50return period (years)

Vol

um

es (

m3

)

1961-1994 2020s 2050s 2080s

Model : Peyron

0

200

400

600

800

2 5 10 50return period (years)

Vol

umes

(m

3)

1961-1994 2020s 2050s 2080s

Model : Watt et al.

0

200

400

600

800

2 5 10 50return period (years)

Vol

umes

(m

3)

1961-1994 2020s 2050s 2080s

HadCM3A2, Model : Watt et al.

0

200

400

600

800

2 5 10 50return period (years)

Vol

umes

(m

3)

1961-1994 2020s 2050s 2080s

35

3.6. Discussion

As the different results illustrate, there are contradictories information provided

by the two different GCM, with opposite consequences. According to the

simulations based on the UK HadCM3, runoff values will decrease, so the

drainage systems would not require to be retrofitted. But, the runoff values

generated from the Canadian CGCM2 tend to predict a potential future

overwhelming of the drainage network. So uncertainties remain on the capacity of

the drainage system to carry out runoff water. These uncertainties seem due to the

inherent uncertainties in the downscaling results using different GCMs (i.e. the

main sources of uncertainties are in the scenarios) Hence the proposed procedure

appears to provide reliable estimations for current climate, but no clear signal can

be exploited for future periods due to the inherent uncertainties of the GCMs.

3.7. Future studies

Even if the procedure proposed in the present study seems to be reliable, further

studies are necessary to address the following issues:

Other GCMs and greenhouse gases emission scenarios should be investigated to provide a wider range of probable runoff evolutions.

The potential impacts of climate change on the scaling behaviour of extreme rainfall processes need to be investigated to avoid the assumption

of stationarity in the rainfall scaling parameters for current and future

periods (one possible approach is to find the linkage between the scaling

parameters and the climatic conditions).

36

4. Regional analysis of rainfall scaling properties

4.1. Introduction

As indicated in Chapter 2, one of the main advantages of the scaling GEV

approach is its ability to determine the extreme precipitation distributions for

different durations based only on a few parameters; that is, the three parameters

of the GEV distribution for a given duration and the two slopes of the rainfall

scaling functions. However, the estimation of these two slopes requires a certain

amount of rainfall data that may not be sufficiently available or that may not exist

for the site of interest. In order to cope with these issues, this section investigates

the regional variability of the rainfall scaling parameters. In addition, as

mentioned in Section 2.7, this regional analysis could be useful in dealing with the

projections of rainfall scaling parameters for future periods in the context of

climate change.

4.2. Regional analysis

4.2.1. Case study using rainfall data in Singapore

The regional analysis is performed using rainfall records available at nine

raingages located on the Singapore Island (1.20 N 103.50 E, 585 km, Figure 30).

The selection of Singapore location for this case study was based on the more

homogeneous climatic conditions over the whole 585-km2 Singapore area as

compared to the high regional climate variability over the large region of Quebec.

Two different sets of data were available for each station in Singapore: the AMP

for 8 durations (15, 30 and 45 minutes; 1, 3, 6, 12 and 24 hours) and the daily

rainfalls (a day is considered as a rainy day in this study if there is any trace of

rainfall recorded). All the data were given in millimetres, with a precision of 0.1

mm for the daily rainfall and 1mm for the AMP. The periods of data availability

differ between each station (Table 7), so the study is performed on the first 30-

year common period, 1972-2001.

37

Stations

Daily Rainfalls Annual Maximum Precipitations

Ama Keng 1961-2007 1960-2007 (1978)2 Changi 1967-2007 1972-2007 Jurong Industrial 1964-2007 1963-2007 Macritche 1961-2007 1960-2007 Paya Lebar 1961-2007 1960-2007 Seletar 1967-2007 1971-2007 Singapore Orchids 1966-2007 1966-2007 (1969, 70, 76) St James 1961-2007 1960-2007 Tengah 1961-2007 1971-2007 Table 7: Periods of data availability for daily rainfalls and for AMPs for eight durations

4.2.2. Data Analysis

4.2.2.1. Monthly repartition of the extreme rainfall occurrences

As expected, the days with the annual maximum daily rainfall depths occur

mostly during the monsoon periods3 (either from November to early February for

the North-East monsoon, or from July to September for the South-East

monsoon). Indeed, as Singapore is located close to the Equator, the Intertropical

Convergence Zone (ITCZ) structure covers the area during the winter month and

favour regular rainfall events (Chia and Foong, 1991). Table 8 shows the regional

variation of the monthly occurrences of AMPs over the whole study region.

2 (years) = data missing for these years

3 Singaporean National Environment Agency, http://app.nea.gov.sg/cms/htdocs/article.asp?pid=1088

Figure 30:: Location of

38

f the nine rainngage stationns in Singapoore

39

Year Ama Keng Changi Juron Ind MacritchePaya Lebar Seletar

Singapore Orchids St James Tengah

1972 9 9 12 12 12 9 12 12 9 1973 3 2 4 2 2 2 2 3 10 1974 9 2 9 12 4 4 11 9 11 1975 10 9 2 7 6 8 11 9 10 1976 10 12 12 11 7 7 12 10 10 1977 5 11 8 11 2 7 10 11 10 1978 12 12 12 12 12 12 12 12 12 1979 12 10 11 10 10 7 3 10 10 1980 1 1 5 1 1 1 1 8 1 1981 7 12 7 5 12 5 5 7 7 1982 11 12 4 11 8 12 12 4 12 1983 5 8 7 7 9 5 12 8 5 1984 3 2 3 3 2 6 3 3 9 1985 5 12 5 5 7 9 5 9 5 1986 3 12 3 9 12 4 3 9 3 1987 8 1 3 1 1 1 1 1 10 1988 5 9 5 11 11 11 7 5 5 1989 11 11 8 11 11 11 12 11 11 1990 6 5 4 10 12 9 5 5 12 1991 12 12 12 12 12 12 12 8 12 1992 12 11 11 11 11 11 11 11 11 1993 9 3 3 10 10 3 9 8 3 1994 3 11 11 11 12 6 4 3 6 1995 2 1 7 8 2 2 7 7 2 1996 8 2 4 8 3 3 2 9 2 1997 6 8 8 12 12 5 3 1 8 1998 8 12 1 7 12 12 6 1 8 1999 8 12 11 1 11 5 10 5 8 2000 4 1 4 4 3 1 1 11 3

Table 8 : Number of months of occurrence of annual maximum daily rainfalls

4.2.3. The scaling GEV parameters

The same method described in the previous chapter was used to examine the

scaling behaviour of the AMP series at each station. As shown in Figures 31 and

32, the AMPs for every station in Singapore indicate a simple scaling behaviour

with two distinct scaling regimes from 15 minutes to 45 minutes and from 45

minutes to one day. Table 9 provides the values of the slopes for the two distinct

rainfall scaling regimes for all nine stations.

40

Figure 31: The scaling behaviour of AMPs for nine Singapore stations

Ama Keng

y = 1016x0.3642

R2 = 0.974

y = 30569x0.5712

R2 = 0.9773

y = 32.513x0.1737

R2 = 0.9698

y = 460.53x1.6427

R2 = 0.9985

y = 59.388x1.0885

R2 = 0.9984

y = 7.716x0.5401

R2 = 0.9983

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1 10 100 1000 10000

duration (min)

NC

MChangi

y = 560.62x0.5248

R2 = 0.9949

y = 12614x0.8319

R2 = 0.9939

y = 24.595x0.2441

R2 = 0.9948

y = 167.59x1.964

R2 = 0.9921

y = 31.269x1.2806

R2 = 0.9918

y = 5.6022x0.6311

R2 = 0.9918

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1 10 100 1000 10000

duration (min)

NC

M

Jurond Industrial

y = 986.18x0.3921

R2 = 0.9768

y = 31505x0.6009

R2 = 0.9757

y = 31.28x0.1914

R2 = 0.978

y = 502.31x1.6658

R2 = 0.9981

y = 62.845x1.1006

R2 = 0.9978

y = 7.8612x0.5471

R2 = 0.9975

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1 10 100 1000 10000

duration (min)

NC

M

Macritche

y = 653.72x0.4624

R2 = 0.9872

y = 11690x0.7849

R2 = 0.9882

y = 28.155x0.2061

R2 = 0.9827

y = 371.17x1.6888

R2 = 0.9931

y = 49.531x1.1329

R2 = 0.993

y = 6.8947x0.5698

R2 = 0.9928

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1 10 100 1000 10000

duration (min)

NC

M

Paya Lebar

y = 562.99x0.5258

R2 = 0.9988

y = 8182x0.9171

R2 = 0.998

y = 26.704x0.2307

R2 = 0.9951

y = 301.1x1.8026

R2 = 0.9951

y = 39.737x1.2263

R2 = 0.9939

y = 6.0479x0.6198

R2 = 0.9927

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1 10 100 1000 10000

duration (min)

NC

M

Seletar

y = 846.13x0.4425

R2 = 0.989

y = 14778x0.7939

R2 = 0.9804

y = 33.01x0.188

R2 = 0.9902

y = 188.83x1.96

R2 = 0.9944

y = 31.795x1.3073

R2 = 0.9941

y = 5.5644x0.6527

R2 = 0.9938

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1 10 100 1000 10000

duration (min)

NC

M

Singapore Orchids

y = 760.57x0.4428

R2 = 0.9804

y = 13538x0.7778

R2 = 0.9628

y = 30.553x0.1934

R2 = 0.9865

y = 363.51x1.7383

R2 = 0.9972

y = 49.416x1.1576

R2 = 0.9973

y = 6.9444x0.5773

R2 = 0.9974

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1 10 100 1000 10000

duration (min)

NC

M

St James

y = 896.48x0.4272

R2 = 0.9881

y = 24779x0.6728

R2 = 0.992

y = 30.809x0.2028

R2 = 0.9813

y = 434.84x1.7155

R2 = 0.9979

y = 56.286x1.1391

R2 = 0.9976

y = 7.4174x0.5681

R2 = 0.9971

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1 10 100 1000 10000

duration (min)

NC

M

Tengah

y = 950.96x0.4192

R2 = 0.9818

y = 25111x0.6865

R2 = 0.9815

y = 32.192x0.1927

R2 = 0.9803

y = 283.08x1.8342

R2 = 0.999

y = 40.956x1.2272

R2 = 0.9984

y = 6.2553x0.6148

R2 = 0.9978

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1 10 100 1000 10000

duration (min)

NC

M

41

Figure 32: Linearity of the slopes of the rainfall scaling behaviour for nine Singapore raingages

Ama Keng

R2 = 1

R2 = 0.99940

0.20.40.60.8

11.21.41.61.8

1 2 3order

beta

Changi

R2 = 0.9998

R2 = 0.9993

0

0.5

1

1.5

2

2.5

1 2 3order

beta

Jurond Industrial

R2 = 1

R2 = 0.9999

00.20.40.60.8

11.21.41.61.8

1 2 3order

beta

Macritche

R2 = 1

R2 = 0.9958

00.20.40.60.8

11.21.41.61.8

1 2 3order

beta

Paya Lebar

R2 = 0.9998

R2 = 0.9935

0

0.5

1

1.5

2

1 2 3order

beta

Seletar

R2 = 1

R2 = 0.99150

0.5

1

1.5

2

2.5

1 2 3order

beta

Singapore Orchids

R2 = 1

R2 = 0.9929

0

0.5

1

1.5

2

1 2 3order

beta

St James

R2 = 1

R2 = 0.9993

0

0.5

1

1.5

2

1 2 3order

beta

Tengah

R2 = 1

R2 = 0.9977

0

0.5

1

1.5

2

1 2 3order

beta

4.2.4. IDF

The IDF c

approach a

distribution

other durati

two identif

rainfall dur

parameters

estimated fr

Figure 3

Table 9: S

F Curves

curves for n

as for Dorv

n for the da

ions were th

fied rainfal

rations can b

and 2 scal

from the rain

33: IDF Curv

Ama KengChangi Jurong IndMacritchePaya LebarSeletar Spore OrchSt James Tengah

Scaling GEV p

nine Singap

val station

ily AMP w

hen comput

ll scaling r

be estimated

ling slopes)

nfall quantil

ves for Ama K

42

be0.0.

ustrial 0.0.

r 0.0.

hids 0.0.0.

parameters f

pore station

in Quebec

were first est

ted using th

regimes. He

d using only

). Finally, t

les given by

Keng station d

eta 1 beta 544 0.182642 0.261551 0.196564 0.233611 0.266653 0.225578 0.22557 0.214613 0.21

for nine Sing

ns were co

. The three

timated, an

he scaling fu

ence, the G

y five param

the IDF cur

y these estim

derived using

2 2 1 6 3 6 5 54

gapore statio

onstructed u

e parameter

nd the GEV

function par

GEV distri

meters (3 da

rves for eac

mated GEV

g Scaling GEV

ns

using the s

rs of the G

parameters

rameters for

butions for

aily AMP G

ch station w

V distribution

V invariance

same

GEV

s for

r the

r all

GEV

were

ns.

e

43

4.3. Methodology

4.3.1. Assessment of relationships between the scaling function slopes and climatic data

As suggested by Yu et al. (2004), the scaling parameters of AMPs might be

related to the number of rainy days at a local site. In order to investigate this

linkage, the correlation coefficients between the scaling slopes and the rainfall

parameters (either the daily AMPs, the mean number of rainy days per year or per

month) were computed. The correlation coefficient is defined as:

,

(22)

in which E is the expected value and the standard deviation. The relationships

are also graphically assessed with plots of the scaling slopes against each of the

14 climatic parameters (average number of rainy days for each month, for the

whole year and average AMP for each station for the whole period).

4.3.2. Estimation of GEV parameters for a partially gauged station

In this case, neither the sub-daily AMP nor the scaling slopes but the 14 rainfall

parameters are known for the study site. The surrounding stations have all these

data. A simple linear regression is established from the surrounding stations,

between both scaling slopes and each of the 14 rainfall parameters. Then, using

the corresponding parameter at the partially gauged site, the scaling slopes at this

site are estimated. The performance of this linear approximation of the scaling

slopes is evaluated using the jackknife method; that is, the scaling slopes at each

site are supposed missing, then estimated from the other eight surrounding

stations, and finally compared with the empirical one at the site, hence, generating

9x2 couples (empirical/estimated) for both scaling slopes.

44

The performance criterion used is the R criterion:

1

(23)

In which is the empirical value of the scaling slope, is the

corresponding estimated value, n the number of stations (here, n=9) and

the variance of the series of the empirical values of scaling slopes.

4.3.3. Estimation of GEV parameters for an ungauged station

In this case, it is assumed that rainfall records (i.e., scaling slopes and rainfall

parameters) are not available at the site of interest. The procedure is similar to the

one used for a partially gauged station, except that the rainfall parameters are

estimated from the ones recorded at the other stations using the formula:

,

,

,

,

(24)

in which is the parameter to be estimated at station i, the corresponding

parameter recorded at station j and , the distance between the stations i and j.

4.4. Results

4.4.1. Relationship between scaling slopes and rainfall parameters

The correlation coefficients, computed for every rainfall parameters are shown in

Table 10 and some typical results of the linear regressions are displayed on

Figures 34, 35 and 36. The scaling slopes are more heavily correlated with the

daily AMP and the number of rainy days in January and February. These results

could be explained by the fact that these 2 months correspond to the North-East

monsoon season, which generates the more intense storms (Chia and Foong,

1991).

So if there

will be an i

inversely, t

are high, th

duration AM

Fig

Table 1

Figure 34

are more ra

intense stor

the scaling

he scaling

MPs.

gure 35: Mea

10: Correlatio

4: Relation be

ainy days in

rm this year

slopes will

slopes are m

an number of

daily AM

annual January

FebruaryMarch April May June July

August September

October NovemberDecember

on coefficient

45

etween daily

n one of the

r, so shorter

l decrease. I

more likely

f rainy days (R

beta10.72-0.08-0.61-0.51-0.38-0.180.000.060.250.28-0.11-0.29-0.10-0.10

ts between sca

AM and the

ese months,

r duration A

In the other

y to be high

RD) in Janua

beta20.89-0.33-0.83-0.76-0.62-0.54-0.34-0.19-0.040.12-0.01-0.61-0.24-0.26

aling slopes a

scaling slope

it is more l

AMP will in

r hand, if th

h too, to in

ary vs. scaling

2 9 3 3 6 2 4 4 9 4 2 1 1 4 6 and hydrolog

es

likely that t

ncrease, an

he daily AM

ndicate sho

g slopes

gic variables.

there

d so

MPs

orter

Fig

4.4.2. Esti

As can be s

estimates fo

gives also c

resulting co

too differen

consequenc

Table 11: P

ure 36: Mean

imation of

seen from t

or both scal

coherent ap

omputed sca

nt compared

ce, the scalin

Performance c

n number of r

scaling slop

the correlati

ling slopes (

pproximate

aling param

d with the ob

ng slopes ca

daily AM

annual JanuaryFebruaryMarch April May June July

AugustSeptember

OctoberNovemberDecember

criterion (Rfor the

46

rainy days (R

pes for par

ion coeffici

(1 and 2).

for the high

meters obtain

bserved one

an be estim

beta1M 0.37

-0.72-0.01-0.15-0.47-0.55-0.67-0.55-0.45-0.66

r -0.42-0.51

r -0.72r -0.56) of the jackke 14 climatic p

RD) in Febru

rtially gaug

ents, the da

The numbe

her duration

ned with the

es (Table 11

mated fairly w

beta20.70-0.470.480.30-0.19-0.27-0.41-0.54-0.46-0.34-0.38-0.18-0.69-0.51

knife methodparameters.

ary vs. scalin

ged stations

aily AMP pr

er of rainy d

n scaling re

e jackknife

1, Figures 3

well with th

2

7

9 7 1 4 6 4 8 8 9 1 for partially

ng slopes

s

rovides the

days in Janu

egime (2).

method are

37 and 38). A

his procedur

y gauged stati

best

uary

The

e not

As a

re.

ions

47

Figure 37: Results of the jackknife method with daily AMP as rainfall parameters for partially gauged site.

Figure 38: Results of the jackknife method with number of rainy days in January as rainfall

parameters for partially gauged site.

48

4.4.3. Estimation of scaling slopes for un-gauged stations

obs com AM obs com wRD Obs com MayRD obs com OctRD Ama Keng 111.6 127.2 13.98% 49.5 48.3 -2.31% 16.1 16.9 4.70% 18.2 18.0 -0.68% Changi 145.9 134.0 -8.14% 31.3 43.5 38.73% 13.3 15.3 15.26% 15.3 15.9 4.11% Jurond Ind 125.2 125.5 0.30% 35.6 47.6 33.85% 14.4 16.4 13.51% 16.8 17.4 3.80% Macritche 125.3 130.6 4.27% 45.7 43.6 -4.57% 15.7 15.6 -0.58% 17.2 16.4 -4.83% Paya Lebar 140.3 133.0 -5.21% 41.9 41.1 -1.95% 14.5 15.4 5.69% 15.1 16.4 8.97% Seletar 130.9 131.1 0.13% 44.2 44.1 -0.19% 16.9 15.6 -7.36% 16.7 16.5 -0.82% Spore Orchad 127.0 126.4 -0.48% 49.8 44.7 -10.15% 18.0 15.9 -11.61% 17.6 17.1 -2.99% St James 130.8 128.1 -2.10% 43.1 43.5 0.79% 13.8 15.7 13.62% 14.7 16.9 14.98% Tengah 127.0 116.4 -8.35% 50.0 47.1 -5.67% 17.2 16.0 -7.06% 18.4 17.8 -3.43%

obs comp aRD obs comp JanRD Obs comp JunRD obs comp NovRD Ama Keng 189.8 193.1 1.78% 15.8 15.9 0.66% 13.6 13.7 1.01% 20.6 20.9 1.38% Changi 150.4 176.9 17.65% 12.4 13.6 10.23% 13.0 13.3 1.76% 19.0 20.1 5.88% Jurond Ind 159.1 187.7 17.95% 15.8 15.3 -3.19% 13.2 13.6 3.31% 19.8 20.5 3.50% Macritche 180.5 176.9 -1.97% 14.5 14.4 -1.02% 14.0 13.3 -4.43% 20.7 20.0 -3.76% Paya Lebar 173.3 172.4 -0.54% 12.7 13.9 9.26% 12.9 13.5 4.94% 19.9 19.9 -0.04% Seletar 183.4 178.8 -2.49% 13.7 14.3 4.24% 13.9 13.5 -3.38% 20.3 20.3 -0.13% Spore Orchad 196.5 180.8 -7.98% 15.8 14.9 -5.69% 14.0 13.6 -3.34% 21.0 20.4 -3.25% St James 161.3 177.9 10.30% 13.9 14.7 5.48% 12.5 13.6 9.15% 18.2 20.4 11.85% Tengah 198.1 184.9 -6.65% 16.1 15.6 -3.22% 13.7 13.5 -1.49% 21.1 20.5 -2.92% obs com sprRD obs com FebRD Obs com JulRD obs com DecRD Ama Keng 49.4 51.5 4.31% 13.3 13.3 -0.23% 13.4 13.5 0.74% 20.4 20.1 -1.38% Changi 36.0 44.1 22.81% 9.7 11.1 14.89% 12.8 13.2 2.83% 19.3 19.2 -0.51% Jurond Ind 41.9 48.8 16.61% 12.6 12.7 1.15% 12.6 13.4 6.53% 18.8 19.9 5.88% Macritche 44.9 45.2 0.68% 11.9 11.8 -0.88% 13.6 13.2 -3.39% 19.3 19.3 -0.01% Paya Lebar 42.1 43.4 3.06% 10.2 11.3 10.88% 13.0 13.3 2.90% 19.0 19.3 1.28% Seletar 47.5 45.2 -4.83% 11.5 11.8 2.58% 13.9 13.3 -4.68% 19.0 19.5 2.52% Spore Orchad 51.9 46.6 -10.21% 13.5 12.2 -9.64% 13.8 13.4 -3.31% 20.5 19.5 -5.01% St James 39.3 45.4 15.44% 10.9 12.1 10.24% 12.1 13.4 10.03% 18.3 19.4 6.29% Tengah 53.3 48.1 -9.72% 13.5 13.0 -3.59% 13.6 13.3 -2.03% 20.3 20.1 -1.21% obs com sumRD obs com MarRD Obs comp AugRD Ama Keng 40.9 41.6 1.58% 16.1 15.9 -0.79% 14.0 14.3 2.61% Changi 39.1 40.8 4.30% 12.6 13.9 10.38% 13.8 14.3 3.61% Jurond Ind 40.2 41.4 2.89% 15.2 15.3 0.75% 13.8 14.3 4.22% Macritche 42.0 40.9 -2.60% 14.8 14.3 -3.82% 14.4 14.4 -0.42% Paya Lebar 40.0 41.2 2.91% 13.2 14.0 6.43% 14.1 14.4 2.04% Seletar 43.1 41.2 -4.40% 14.2 14.5 1.65% 15.2 14.4 -5.01% Spore Orchad 43.5 41.3 -5.18% 16.0 14.9 -6.48% 15.7 14.3 -8.95% St James 37.7 41.4 9.81% 12.3 14.8 20.92% 13.1 14.4 9.54% Tengah 41.6 41.0 -1.34% 16.2 15.7 -3.04% 14.3 14.1 -1.13% obs comp falRD obs comp AprRD Obs comp SepRD Ama Keng 53.7 53.4 -0.69% 18.8 18.3 -2.68% 14.9 14.9 -0.27% Changi 43.3 50.5 16.73% 14.9 16.1 7.83% 14.2 14.8 4.23% Jurond Ind 44.8 52.8 17.96% 16.2 17.7 9.41% 14.2 14.9 5.25% Macritche 53.0 50.1 -5.49% 16.6 16.6 -0.33% 15.1 14.7 -2.24% Paya Lebar 49.7 49.3 -0.97% 15.3 16.4 7.09% 14.7 14.6 -0.66% Seletar 51.6 51.1 -1.01% 17.2 16.6 -3.02% 14.6 15.0 2.67% Spore Orchad 54.9 51.2 -6.80% 18.6 17.2 -7.66% 16.3 14.7 -9.62% St James 46.7 50.8 8.85% 14.4 16.9 17.33% 13.7 14.9 8.57% Tengah 54.4 52.3 -3.75% 18.7 18.2 -2.77% 14.9 14.9 0.30%

Table 12: Performance of the parameter estimation method (com=computed values and obs=observed values, the percentage being the difference between these values)

49

The proposed formula (Equation 24) for estimating rainfall parameters at an

ungaged site gives results comparable to the observed values (Table 12). The

stations with large discrepancies (Changi and Jurong Industrial Waterworks) are

the ones with a large distance between them. In addition, their geographical

situations near the coast of the island could explain these discrepancies as well

since they have fewer neighbours with similar coastal effects, they have lower

probability of having other stations with similar hydrologic features. Nevertheless,

even theses estimates are fairly similar with the observed values.

On the basis of these good estimates, the slope of the second scaling regime is

fairly well predicted at ungaged stations when the linear regression is based on the

number of rainy days either in January or in February (Table 13 and Figure 39).

However, the method is not able to do better than a random process to evaluate 1 (R

Figure 39: R

4.5.

The presen

slopes and

study have

slopes base

interest has

local inform

(over than

improve th

without rai

estimation

al., 2004) th

Results of the

Discussion

nt study ai

some rain

indicated th

ed on infor

s some info

mation, only

45 minute

he estimatio

infall recor

methods or

hat could pr

Jacknife met

n

ims at inve

nfall param

hat it is po

rmation fro

ormation re

y the slope o

es) can be

on of the

ds further

r to find oth

rovide a stro

50

thod based onthe 9 statio

estigating th

meters using

ossible to es

om surround

egarding the

of the temp

reasonably

scaling fun

studies are

her rainfall

onger relatio

n the numberons

he relations

g a regiona

stimate fair

ding station

e rainfall

oral downs

y estimate

nctions for

needed to

l parameters

on with the

r of rainy day

ships betwe

l analysis.

rly well the

ns as long

parameters.

caling for lo

ed. Howeve

location w

o consider o

s (see, for

scaling slop

ys in January

een the sca

Results of

rainfall sca

as the sit

. If there is

onger durat

er, in orde

with limited

other nonlin

instance, Y

pes.

y for

aling

this

aling

e of

s no

tions

er to

d or

near

Yu et

51

5. Conclusions

The main objective of this stud